Nonisomorphic two-dimensional algebraically defined graphs over R R

For $f:R^2\to R$, let ${\Gamma }_R\left(f\right)$ be a two-dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of $R^2$ and two vertices $\left(a,a_2\right)$ and $\left[x,x_2\right]$ are adjacent if and only if $a_2+x_2=f\left(a,x\right)$. It is known that ${\Gamma }_R\left(XY\right)$ has girth 6 and can be extended to the point-line incidence graph of the classical real projective plane. However, it was unknown whether there exists $f\in R\left[X,Y\right]$ such that ${\Gamma }_R\left(f\right)$ has girth 6 and is nonisomorphic to ${\Gamma }_R\left(XY\right)$. This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of ${\Gamma }_R\left(f\right)$ for families of bivariate functions $f$.